The photoacoustic (PA), or more precisely, the photo-thermo-acoustic (PTA) effect is the process of light being absorbed by a material, creating a temperature change followed by a localized volume expansion leading to the generation of acoustic waves. In addition to obvious applications in the area of sub-surface depth profilometry of defects in materials (see Mandelis in Progress in Photothermal and Photoacoustic Science and Technology, North-Holland, New York, 1992), there have been many advances in applying photoacoustic phenomena to soft tissue imaging, cancerous lesion detection, and sub-dermal depth profilometry in the last decade. In recent years, application of laser photoacoustic to soft tissue imaging, cancerous lesion detection, and sub-dermal depth profilometry has enjoyed very rapid development (see A. A. Oraevsky in Biomedical Optoacoustics, Proc. SPIE Vol. 3916 and Vol. 4256 and A. J. Welch and M. C. van Gemert in Tissue Optical Properties and Laser-Tissue Interactions, AIP, New York, 1995), becoming the object of broader attention by the biomedical optics community (see Biophotonics International, September/October 2000, pp. 40-45). This is so because PA detection has shown concrete promise of depth profilometric imaging in turbid media at depths significantly larger than accessible by purely optical methodologies (See V. G. Andreev, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. Aleynikov, Y. Z. Zhulina, R. D. Fleming, and A. A. Oraevsky in Biomedical Optoacoustics, Proc. SPIE Vol. 3916). In state-of-the-art laser PA instrumentation and measurement systems in turbid media, as developed by some of the major research groups in this field (see V. G. Andreev, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. Aleynikov, Y. Z. Zhulina, R. D. Fleming, and A. A. Oraevsky in Biomedical Optoacoustics, Proc. SPIE Vol. 3916; G. A. Hoelen, R. G. M. Kolkman, M. Letteboer, R. Berendsen and F. F. de Mul in Optical Tomography and Spectroscopy of Tissue III, Proc. SPIE Vol. 3597, pp 336; P. C. Beard and T. N. Mills in Proc. SPIE Vol. 3916, pp. 100; L. H. Wang, S. L. Jacques and X. Zhao in Opt. Lett. 20, 629, 1995; G. Yao and L. H. Wang in Appl. Opt. 39, pp. 659), a pulsed laser has always been the source of choice for optical generation of PA signal. The major reasons for this choice are two: a) Following optical absorption of a short laser pulse by turbid tissue, optical-to-thermal energy conversion and localized PTA volume expansion, an acoustic transient received within approximately 1 μs after the end of the laser pulse is essentially thermally adiabatic: it carries information about the thermal shape of the absorber, which substantially coincides with its geometric shape before any significant heat conduction can deform the image at later times (see A. A. Karabutov, N. B. Podymova and V. S. Letokhov in Appl. Opt. 34, pp. 1484); b) In pulsed photoacoustics, a large amount of the available energy is imparted to the Fourier spectral components of the PTA signal response, which correspond to the early-times (or high frequencies) after the arrival of the acoustic pulse at the transducer, thus yielding acceptable signal-to-noise ratios under co-added transient pulse detection (see A. Mandelis in Rev. Sci. Instrum. 65, pp. 3309). Pulsed PTA detection, however, presents disadvantages in terms of laser jitter noise, acoustic and thermal noise within the wide bandwidth of the transducer, hard-to-control depth localization of the contrast-generating sub-surface features, as well as strong background signals from sound scattering tissues. These mechanisms tend to limit system detectivity and signal-to-noise ratio (SNR) and amount to important limitations because they may seriously compromise the contributions to the signal of contrast-generating subsurface features and thus limit the ability to monitor nascent and small size tumors by the PTA technique. In addition, very large pulsed-laser peak fluences incident on living tissue may have detrimental effects and for this reason average pulse energies are very low (<5 mJ) resulting in poor SNR. Besides, it is difficult to construct linear, low-noise detection systems for wide range of pulsed amplitudes, a common requirement for patient-specific diagnostics and laser therapy. Normally, acoustic responses from turbid media are time-gated and Fourier transformed into the frequency domain in order to determine and match the peak response of the transducer with the frequency contents of the PTA signal (see A. Oraevsky and A. Karabutov in Biomedical Optoacoustics, Proc. SPIE Vol. 3916, pp 228). Quantitatively, tissue inhomogeneity parameter measurements are derived from the peak of the frequency spectrum of the transformed signal.
Frequency-domain (FD) PTA methodologies can offer alternative detection and imaging schemes with concrete advantages over pulsed laser photoacoustics. These advantages include: a) Low fluence of the harmonic or frequency-swept (chirped) laser modulation, with the concomitant advantage of a much higher tissue damage threshold. A combination of harmonically modulated and chirped detection methodologies (see A. Mandelis in Rev. Sci. Instrum. 65, pp. 3309) can overcome the possible disadvantage of lower signal levels under single-frequency harmonic modulation at high, thermally adiabatic, frequencies (˜ MHz), while retaining the speed and wide temporal range of pulsed laser responses. The superior signal-to-noise ratio of the ultra-narrow lock-in amplifier band-pass filter can offset much of the SNR deterioration at MHz frequencies. Frequency chirps, may also recover the strength of the high-frequency Fourier components through fast-Fourier transformation of the frequency-domain transfer function to time-domain impulse-response, thus matching the major advantage of pulsed-laser excitation; b) Depth profilometry over very wide range of frequencies. The depth range in turbid media depends on the acoustic velocity and the optical extinction coefficient at the probe wavelength; c) Possible parallel multi-channel lock-in signal processing and image generation in quasi-real time (see D. Fournier, F. Charbonnier and A. C. Boccara in French Patent 2666, 1993 and J. Selb, S. Leveque-Fort, L. Pottier and C. Boccara in Biomedical Optoacoustics II, Proc. SPIE Vol. 4256); d) A substantially wider signal dynamic range through use of lock-in filtering; and e) A simple instrumental normalization procedure through division with a reference signal in the frequency-domain, as opposed to non-trivial deconvolution in the time-domain, especially with highly non-linear (e.g. resonant) ultrasonic transducers. Yet, FD and/or hybrid methods have not been pursued historically in biomedical PTA imaging.
Theory: PTA Wave Generation from a Turbid Solid Immersed in a Fluid
FIG. 1 shows the geometry used for the one-dimensional mathematical model. The configuration closely corresponds to the experimental geometry. It contains three coupled layers: the top and bottom layers are composed of a fluid with the middle layer composed of a solid. The top layer is assumed to be semi-infinite fluid and occupies the spatial region −∞<z≦−L. It has density ρf and speed of sound cf. The solid layer has thickness L, density ρs, speed of sound cs, specific heat at constant pressure CPs, optical absorption coefficient at the laser wavelength μa, optical scattering coefficient μs, bulk modulus Ks and isobaric volume expansion coefficient βs. The bottom layer extends from 0≦z<∞. The reason for not considering the finite thickness of the bottom layer is that in our experiments no reflections from the fluid-container interface are observed. However, this feature can be readily added as a straightforward extension to the mathematical theory, as it is easily understood by those skilled in the art.
An analytical solution of the coupled PTA problem in the form of spectral integrals can be obtained by converting the time-domain equations to their frequency-domain counterparts using Fourier transformations (FT) (see A. Karabutov and V. Gusev in Laser Optoacoustics, AIP Press, New York, 1993 and A. Mandelis, N. Baddour, Y. Cai and R. Walmsley in J. Opt. Soc. Am. B (in press)). For a harmonic optical source, the Fourier transform of the radiative transfer equation yields Eq. (1), which is satisfied by the diffuse photon density wave (DPDW, or diffuse radiant energy fluence rate) field (see T. J. Farrell, M. S. Patterson and B. Wilson in Med. Phys. 19, pp. 879), ψd [Wm−2]:
                                                                                          ∂                  2                                                  ∂                                      z                    2                                                              ⁢                                                ψ                  d                                ⁡                                  (                                      z                    ,                    ω                                    )                                                      -                                          σ                p                2                            ⁢                                                ψ                  d                                ⁡                                  (                                      z                    ,                    ω                                    )                                                              =                                    I              0              ′                        ⁢                          ⅇ                              -                                                      μ                    t                                    ⁡                                      (                                          z                      +                      L                                        )                                                                                      ,                              -            L                    ≤          z          ≤          0                                    (        1        )            
Here a source strength depth distribution is assumed that decreases exponentially into the turbid medium (Bouguet's law) with total attenuation (extinction) coefficient:
                                          μ            t                    =                                    μ              s                        +                          μ              a                                      ,                                  ⁢        Also        ,                            (        2        )                                                      I            0            ′                    =                                                    -                                                                            I                      0                                        ⁢                                          μ                      s                                                        D                                            ⁢                              (                                                                            μ                      t                                        +                                          g                      ⁢                                                                                          ⁢                                              μ                        a                                                                                                                        μ                      t                                        -                                          g                      ⁢                                                                                          ⁢                                              μ                        s                                                                                            )                            ⁢                                                          ⁢              and              ⁢                                                          ⁢              D                        =                          1                              3                ⁡                                  [                                                            μ                      a                                        +                                                                  (                                                  1                          -                          g                                                )                                            ⁢                                              μ                        s                                                                              ]                                                                    ,                            (        3        )            where I0 is the laser fluence, g is the mean cosine of the scattering function of the photon field over all spatial directions described by the solid angle. In view of the almost entirely forward scattering of photons in tissue, g values range between 0.6 and 0.98 (see W. M. Star and J. P. A. Marijnissen in J. Photochem. Photobiol., B 1, 149).
The complex diffuse-photon wave number is defined as (see A. Mandelis in Diffusion-Wave Fields: Mathematical Methods and Green Functions, Springer-Verlag, New York, 2001, Chap. 10, pp. 663-708):
                                          σ            p                    =                                                                      1                  -                                      ⅈ                    ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                                          τ                      a                                                                                                            D                    eff                                    ⁢                                      τ                    a                                                                        ⁢                                                  ⁢            where                          ,                                  ⁢                              D            eff                    =                      vD            =                          v                              3                ⁡                                  [                                                            μ                      a                                        +                                                                  (                                                  1                          -                          g                                                )                                            ⁢                                              μ                        s                                                                              ]                                                                    ,                              τ            a                    =                                                    (                                  v                  ⁢                                                                          ⁢                                      μ                    a                                                  )                                            -                1                                      .                                              (        4        )            
Here ν is the speed of light (≈1010 cm/s for light propagating in turbid media); D is the optical diffusion coefficient, in units of length. The general solution of Eq. (1) is:ψd(z,ω)=A1eσp(z+L)+A2e−σp(z+L)+Be−μ1(z+L)  (5)
Constant B can be determined as:
                    B        =                              I            0            ′                                              μ              t              2                        -                          σ              p              2                                                          (        6        )            
Constants A1, and A2 can be solved using the boundary conditions for the DPDW [20]:
                                                                        ψ                d                            ⁡                              (                                                      -                    L                                    ,                  ω                                )                                      -                          A              ⁢                                                ∂                                                                                                          ∂                  z                                            ⁢                                                ψ                  d                                ⁡                                  (                                                            -                      L                                        ,                    ω                                    )                                                              =                                    -              3                        ⁢                          μ              s                        ⁢                          gAI              0                                      ⁢                                  ⁢                                                            ψ                d                            ⁡                              (                                  0                  ,                  ω                                )                                      -                          A              ⁢                                                ∂                                                                                                          ∂                  z                                            ⁢                                                ψ                  d                                ⁡                                  (                                      0                    ,                    ω                                    )                                                              =                      3            ⁢                          μ              s                        ⁢                          gAI              0                        ⁢                          ⅇ                              -                                                      μ                    t                                    ⁡                                      (                                          z                      +                      L                                        )                                                                                                          (        7        )            where
  A  =            2      ⁢              D        ⁡                  (                                    1              +                              r                21                                                    1              -                              r                21                                              )                      ≡          2      ⁢      D      ⁢                          ⁢              ξ        ·                  r          21                    is the internal reflectance, defined as the ratio of the upward-to-downward hemispherical diffuse optical fluxes at the boundary.
Therefore,
                                                        A              2                        =                                                                                                      F                      1                                        ⁡                                          (                                              1                        +                                                  A                          ⁢                                                                                                          ⁢                                                      σ                            p                                                                                              )                                                        ⁢                                      ⅇ                                                                  σ                        p                                            ⁢                      L                                                                      -                                                                            F                      2                                        ⁡                                          (                                              1                        -                                                  A                          ⁢                                                                                                          ⁢                                                      σ                            p                                                                                              )                                                        ⁢                                      ⅇ                                                                  -                                                  σ                          p                                                                    ⁢                      L                                                                                                                                                              (                                              1                        +                                                  A                          ⁢                                                                                                          ⁢                                                      σ                            p                                                                                              )                                        2                                    ⁢                                      ⅇ                                                                  σ                        p                                            ⁢                      L                                                                      -                                                                            (                                              1                        -                                                  A                          ⁢                                                                                                          ⁢                                                      σ                            p                                                                                              )                                        2                                    ⁢                                      ⅇ                                                                  -                                                  σ                          p                                                                    ⁢                      L                                                                                                    ;                ⁢                                  ⁢                                            A              1                        =                                                            F                  1                                -                                                      A                    2                                    ⁡                                      (                                          1                      +                                              A                        ⁢                                                                                                  ⁢                                                  σ                          p                                                                                      )                                                                              (                                  1                  +                                      A                    ⁢                                                                                  ⁢                                          σ                      p                                                                      )                                              ;                ⁢                                  ⁢                                            F              1                        =                                          3                ⁢                                  μ                  s                                ⁢                                  gAI                  0                                            -                              B                ⁡                                  (                                      1                    +                                          A                      ⁢                                                                                          ⁢                                              μ                        t                                                                              )                                                              ;                ⁢                                  ⁢                  F          2                =                              3            ⁢                          μ              s                        ⁢                          gAI              0                        ⁢                          ⅇ                                                -                                      μ                    t                                                  ⁢                L                                              -                                    B              ⁡                              (                                  1                  -                                      A                    ⁢                                                                                  ⁢                                          μ                      t                                                                      )                                      ⁢                                          ⅇ                                                      -                                          μ                      t                                                        ⁢                  L                                            .                                                          (        8        )            
To complete the solution, the coherent photon-density field, ψc=I0e−μ1(z+L), must be added to the diffuse-photon-density distribution, ψd. The total photon density field ψt=ψdψc.
In frequency domain, the thermal-wave equation can be written as:
                                                                                          ∂                  2                                                  ∂                                      z                    2                                                              ⁢                                                θ                  s                                ⁡                                  (                                      z                    ,                    ω                                    )                                                      -                                          (                                                      ⅈ                    ⁢                                                                                  ⁢                    ω                                                        α                    s                                                  )                            ⁢                                                θ                  s                                ⁡                                  (                                      z                    ,                    ω                                    )                                                              =                                    -                                                                    η                    NR                                    ⁢                                      μ                    a                                                                    λ                  s                                                      ⁢                                          ψ                t                            ⁡                              (                                  z                  ,                  ω                                )                                                    ,                            (        9        )            where θs (z,ω) is the thermoelastic temperature rise above ambient. αs and λs are, respectively, the thermal diffusivity and conductivity of the solid medium. The general solution of Equation (9) is:θs(z,ω)=C1e−σs(z+L)+C1beσs(z+L)+C2eσp(z+L)+C3e−σp(z+L)+C4e−μt(z+L).   (10)
Constants C2, C3, C4 can be solved as:
                                                        C              2                        =                                                            η                  NR                                ⁢                                  μ                  a                                ⁢                                  A                  1                                                                              λ                  s                                ⁡                                  (                                                            σ                      s                      2                                        -                                          σ                      p                      2                                                        )                                                              ;                ⁢                                  ⁢                                            C              3                        =                                                            η                  NR                                ⁢                                  μ                  a                                ⁢                                  A                  2                                                                              λ                  s                                ⁡                                  (                                                            σ                      s                      2                                        -                                          σ                      p                      2                                                        )                                                              ;                ⁢                                  ⁢                                            C              4                        =                                                            η                  NR                                ⁢                                                      μ                    a                                    ⁡                                      (                                          B                      +                      1                                        )                                                                                                λ                  s                                ⁡                                  (                                                            σ                      s                      2                                        -                                          μ                      t                      2                                                        )                                                              ;                                    (        11        )            
In the fluid z≦−L and z≧0, the temperature field can be written, respectively, as:θf(z,ω)=Ceσf(z+L)z≦−L. θf(z,ω)=Cbe−σfzz≧0.   (12)
Constants C1, C1b, C, Cb can be solved using the boundary conditions of thermal continuity at the fluid/solid interfaces:
                                                                        θ                f                            ⁡                              (                                                      -                    L                                    ,                  ω                                )                                      =                                          θ                s                            ⁡                              (                                                      -                    L                                    ,                  ω                                )                                              ,                                          ⁢                                                    λ                s                            ⁢                                                ∂                                                                                                          ∂                  z                                            ⁢                                                θ                  s                                ⁡                                  (                                                            -                      L                                        ,                    ω                                    )                                                      =                                          λ                f                            ⁢                                                ∂                                                                                                          ∂                  z                                            ⁢                                                θ                  f                                ⁡                                  (                                                            -                      L                                        ,                    ω                                    )                                                              ,                                          ⁢                                                    θ                f                            ⁡                              (                                  0                  ,                  ω                                )                                      =                                          θ                s                            ⁡                              (                                  0                  ,                  ω                                )                                                    ⁢                                  ⁢                                            λ              s                        ⁢                                          ∂                                                                                              ∂                z                                      ⁢                                          θ                s                            ⁡                              (                                  0                  ,                  ω                                )                                              =                                    λ              f                        ⁢                                          ∂                                                                                              ∂                z                                      ⁢                                          θ                f                            ⁡                              (                                  0                  ,                  ω                                )                                                                        (        13        )                                Therefore        ,                            (        14        )                                                      C            1                    =                                                                                                                                                                  -                                                      λ                            f                                                                          ⁢                                                                              σ                            f                                                    ⁡                                                      (                                                                                          C                                2                                                            +                                                              C                                3                                                            +                                                              C                                4                                                                                      )                                                                                              +                                                                                                                                                                                    λ                          s                                                ⁡                                                  (                                                                                                                    σ                                p                                                            ⁢                                                              C                                2                                                                                      -                                                                                          σ                                p                                                            ⁢                                                              C                                3                                                                                      -                                                                                          μ                                t                                                            ⁢                                                              C                                4                                                                                                              )                                                                    -                                                                                  ⁢                                                                                          C                      rhs                                        ⁡                                          (                                                                                                    λ                            f                                                    ⁢                                                      σ                            f                                                                          -                                                                              λ                            s                                                    ⁢                                                      σ                            s                                                                                              )                                                        ⁢                                      ⅇ                                                                  -                                                  σ                          s                                                                    ⁢                      L                                                                                        (                                                                                    λ                        f                                            ⁢                                              σ                        f                                                              +                                                                  λ                        s                                            ⁢                                              σ                        s                                                                              )                                                                                    (                                                                            λ                      f                                        ⁢                                          σ                      f                                                        +                                                            λ                      s                                        ⁢                                          σ                      s                                                                      )                            -                                                                                          (                                                                                                    λ                            f                                                    ⁢                                                      σ                            f                                                                          +                                                                              λ                            s                                                    ⁢                                                      σ                            s                                                                                              )                                        2                                    ⁢                                      ⅇ                                                                  -                        2                                            ⁢                                              σ                        s                                            ⁢                      L                                                                                        (                                                                                    λ                        f                                            ⁢                                              σ                        f                                                              +                                                                  λ                        s                                            ⁢                                              σ                        s                                                                              )                                                                    ;                                                                      C                      1            ⁢            b                          =                                            C              rhs                        -                                          (                                                                            λ                      f                                        ⁢                                          σ                      f                                                        -                                                            λ                      s                                        ⁢                                          σ                      s                                                                      )                            ⁢                              ⅇ                                                      -                                          σ                      s                                                        ⁢                  L                                            ⁢                              C                1                                                                        (                                                                    λ                    f                                    ⁢                                      σ                    f                                                  +                                                      λ                    s                                    ⁢                                      σ                    s                                                              )                        ⁢                          ⅇ                                                σ                  s                                ⁢                L                                                                                                                  C          rhs                =                                            λ              s                        ⁡                          (                                                                    σ                    p                                    ⁢                                      C                    3                                    ⁢                                      ⅇ                                                                  -                                                  σ                          p                                                                    ⁢                      L                                                                      +                                                      μ                    t                                    ⁢                                      C                    4                                    ⁢                                      ⅇ                                                                  -                                                  μ                          t                                                                    ⁢                      L                                                                      -                                                      σ                    p                                    ⁢                                      C                    2                                    ⁢                                      ⅇ                                                                  σ                        p                                            ⁢                      L                                                                                  )                                -                                    λ              f                        ⁢                                          σ                f                            ⁡                              (                                                                            C                      3                                        ⁢                                          ⅇ                                                                        -                                                      σ                            p                                                                          ⁢                        L                                                                              +                                                            C                      4                                        ⁢                                          ⅇ                                                                        -                                                      μ                            t                                                                          ⁢                        L                                                                              +                                                            C                      2                                        ⁢                                          ⅇ                                                                        σ                          p                                                ⁢                        L                                                                                            )                                                                                    
By introducing in the solid a particle/molecule displacement potential, φs (z,ω), the coupled wave equations in the solid and fluid can be easily solved. The displacement potential is related to the magnitude of the one-dimensional displacement vector, Us (z,ω), as:
                                                        U              s                        ⁡                          (                              z                ,                ω                            )                                =                                    ∂                              ∂                z                                      ⁢                                          ϕ                s                            ⁡                              (                                  z                  ,                  ω                                )                                                    ,                              -            L                    ≤          z          ≤          0.                                    (        15        )            
Due to laser PTA excitation by a large spot-size laser beam, further expanded by intra-solid optical scattering, only longitudinal waves are assumed to propagate in an isotropic solid. This assumption allows the use of the Helmholtz equation which is satisfied by the displacement potential, φs:
                                                                                          ⅆ                  2                                                  ⅆ                                      z                    2                                                              ⁢                                                ϕ                  s                                ⁡                                  (                                      z                    ,                    ω                                    )                                                      +                                          k                s                2                            ⁢                                                ϕ                  s                                ⁡                                  (                                      z                    ,                    ω                                    )                                                              =                                    (                                                                    K                    s                                    ⁢                                      β                    s                                                                                        ρ                    s                                    ⁢                                      c                    s                    2                                                              )                        ⁢                                          θ                s                            ⁡                              (                                  z                  ,                  ω                                )                                                    ,                            (        16        )            where ks=ω/cs is the acoustic wavenumber in the solid for small-amplitude acoustic perturbations. The general solution to this equation is:φs(z,ω)=G1eiksz+G2eiksz+G3e−σs(z+L)+G4eσp(z+L)+G5e−σp(z+L)+G6e−μt(z+L)+G9eσs(z+L)  (17)
Constants G3, G4, G5 and G6 are found to be:
                                                        G              3                        =                                                            K                  s                                ⁢                                  β                  s                                ⁢                                  C                  1                                                                              ρ                  s                                ⁢                                                      c                    s                    2                                    ⁡                                      (                                                                  σ                        s                        2                                            +                                              k                        s                        2                                                              )                                                                                ,                                          ⁢                                    G              4                        =                                                            K                  s                                ⁢                                  β                  s                                ⁢                                  C                  2                                                                              ρ                  s                                ⁢                                                      c                    s                    2                                    ⁡                                      (                                                                  σ                        p                        2                                            +                                              k                        s                        2                                                              )                                                                                ,                                          ⁢                                    G              5                        =                                                            K                  s                                ⁢                                  β                  s                                ⁢                                  C                  3                                                                              ρ                  s                                ⁢                                                      c                    s                    2                                    ⁡                                      (                                                                  σ                        p                        2                                            +                                              k                        s                        2                                                              )                                                                                      ⁢                                  ⁢                              G            6                    =                                                    K                s                            ⁢                              β                s                            ⁢                              C                4                                                                    ρ                s                            ⁢                                                c                  s                  2                                ⁡                                  (                                                            μ                      t                      2                                        +                                          k                      s                      2                                                        )                                                                    ⁢                                  ⁢                              G            9                    =                                                    K                s                            ⁢                              β                s                            ⁢                              C                                  1                  ⁢                  b                                                                                    ρ                s                            ⁢                                                c                  s                  2                                ⁡                                  (                                                            σ                      s                      2                                        +                                          k                      s                      2                                                        )                                                                                        (        18        )            
Inside the fluid, since wave sources are of a potential nature, liquid motion will be potential-driven motion. By introducing a scalar potential of the velocity field,
                                          v            ⁡                          (                              z                ,                ω                            )                                =                                    ∂                              ∂                z                                      ⁢                                          ψ                                  f                  i                                            ⁡                              (                                  z                  ,                  ω                                )                                                    ,                              -            ∞                    <          z          ≤                                    -                        ⁢            L                          ,                  0          ≤          z          <          ∞                ,                            (        19        )            where the subscript i=1, 2 indicates the top and bottom fluid, respectively, one can obtain the photo-thermo-acoustic wave equation (Eq. (20)) for a non-viscous fluid:
                                                                        ⅆ                2                                            ⅆ                                  z                  2                                                      ⁢                                          ψ                                  f                  i                                            ⁡                              (                                  z                  ,                  ω                                )                                              +                                    k              f              2                        ⁢                                          ψ                                  f                  i                                            ⁡                              (                                  z                  ,                  ω                                )                                                    =        0                            (        20        )            where kf=ω/cf is the wavenumber for small-amplitude acoustic perturbations in the fluid.
The small-amplitude pressure change in the fluid is related to the velocity potential, Ψfi by:P(z,ω)=−iωρfψfi(z,ω).   (21)
The general solutions to Equation (20) can be written as:ψf1(z,ω)=G7eikf(z+L), −∞<z≦−L ψf2(z,ω)=G8eikfz. 0≦z<∞  (22)
The constants (G1, G2, G7, G8) in equations (17) and (22) can be determined through the boundary conditions of stress and velocity continuity at the two interfaces z=0, −L.
                                                                        ρ                s                            ⁢                              c                s                2                            ⁢                                                ⅆ                  2                                                  ⅆ                                      z                    2                                                              ⁢                                                ϕ                  s                                ⁡                                  (                                      0                    ,                    ω                                    )                                                      -                                          K                s                            ⁢                              β                s                            ⁢                                                θ                  s                                ⁡                                  (                                      0                    ,                    ω                                    )                                                              =                                    -                              P                ⁡                                  (                                      0                    ,                    ω                                    )                                                      =                                          ⅈωρ                f                            ⁢                                                ψ                  f2                                ⁡                                  (                                      0                    ,                    ω                                    )                                                                    ,                                                            ρ                s                            ⁢                              c                s                2                            ⁢                                                ⅆ                  2                                                  ⅆ                                      z                    2                                                              ⁢                                                ϕ                  s                                ⁡                                  (                                                            -                      L                                        ,                    ω                                    )                                                      -                                          K                s                            ⁢                              β                s                            ⁢                                                θ                  s                                ⁡                                  (                                                            -                      L                                        ,                    ω                                    )                                                              =                                    P              ⁡                              (                                                      -                    L                                    ,                  ω                                )                                      =                                          ⅈωρ                f                            ⁢                                                ψ                  f1                                ⁡                                  (                                                            -                      L                                        ,                    ω                                    )                                                                    ,                              ⅈω            ⁢                          ⅆ                              ⅆ                z                                      ⁢                                          ϕ                s                            ⁡                              (                                  0                  ,                  ω                                )                                              =                                    ⅆ                              ⅆ                z                                      ⁢                                          ψ                f2                            ⁡                              (                                  0                  ,                  ω                                )                                                    ,                ⁢                              ⅈω            ⁢                          ⅆ                              ⅆ                z                                      ⁢                                          ϕ                s                            ⁡                              (                                                      -                    L                                    ,                  ω                                )                                              =                                    ⅆ                              ⅆ                z                                      ⁢                                                            ψ                  f1                                ⁡                                  (                                                            -                      L                                        ,                    ω                                    )                                            .                                                          (        23        )            
Substituting the displacement potential, temperature field, and velocity potentials into the boundary conditions, Eq. (23) can be written as:
                                          [                                                                                A                    11                                                                                        A                    12                                                                                        A                    13                                                                                        A                    14                                                                                                                    A                    21                                                                                        A                    22                                                                                        A                    23                                                                                        A                    24                                                                                                                    A                    31                                                                                        A                    32                                                                                        A                    33                                                                                        A                    34                                                                                                                    A                    41                                                                                        A                    42                                                                                        A                    43                                                                                        A                    44                                                                        ]                    ⁡                      [                                                                                G                    1                                                                                                                    G                    2                                                                                                                    G                    7                                                                                                                    G                    8                                                                        ]                          =                              [                                                                                H                    1                                                                                                                    H                    2                                                                                                                    H                    3                                                                                                                    H                    4                                                                        ]                    .                                    (        24        )                        where                                                                            H            1                    =                                                    K                s                            ⁢                                                β                  s                                ⁡                                  (                                                            C                      1                                        +                                          C                      2                                        +                                          C                      3                                        +                                          C                      4                                                        )                                                      -                                          σ                s                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              G                3                                      -                                          σ                p                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              G                4                                      -                                          σ                p                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              G                5                                      -                                          μ                t                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              G                6                                                    ;                                                                                  H            2                    =                                                    K                s                            ⁢                                                β                  s                                ⁡                                  (                                                                                    C                        1                                            ⁢                                              ⅇ                                                                              -                                                          σ                              s                                                                                ⁢                          L                                                                                      +                                                                  C                        2                                            ⁢                                              ⅇ                                                                              σ                            p                                                    ⁢                          L                                                                                      +                                                                  C                        3                                            ⁢                                              ⅇ                                                                              -                                                          σ                              p                                                                                ⁢                          L                                                                                      +                                                                  C                        4                                            ⁢                                              ⅇ                                                                              -                                                          μ                              t                                                                                ⁢                          L                                                                                                      )                                                      -                                          σ                s                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              ⅇ                                                      -                                          σ                      s                                                        ⁢                  L                                            ⁢                              G                3                                      -                                          σ                p                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              ⅇ                                                      σ                    p                                    ⁢                  L                                            ⁢                              G                4                                      -                                          σ                p                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              ⅇ                                                      -                                          σ                      p                                                        ⁢                  L                                            ⁢                              G                5                                      -                                          μ                t                2                            ⁢                              ρ                s                            ⁢                              c                s                2                            ⁢                              ⅇ                                                      -                                          μ                      t                                                        ⁢                  L                                            ⁢                              G                6                                                    ;                                                                                  H            3                    =                      ⅈω            ⁡                          (                                                                    σ                    s                                    ⁢                                      G                    3                                                  -                                                      σ                    p                                    ⁢                                      G                    4                                                  +                                                      σ                    p                                    ⁢                                      G                    5                                                  +                                                      μ                    t                                    ⁢                                      G                    6                                                              )                                      ;                                                                                  H            4                    =                      ⅈω            ⁢                          (                                                                    σ                    s                                    ⁢                                      G                    3                                    ⁢                                      ⅇ                                                                  -                                                  σ                          s                                                                    ⁢                      L                                                                      -                                                      σ                    p                                    ⁢                                      G                    4                                    ⁢                                      ⅇ                                                                  σ                        p                                            ⁢                      L                                                                      +                                                      σ                    p                                    ⁢                                      G                    5                                    ⁢                                      ⅇ                                                                  -                                                  σ                          p                                                                    ⁢                      L                                                                      +                                                      μ                    t                                    ⁢                                      G                    6                                    ⁢                                      ⅇ                                                                  -                                                  μ                          t                                                                    ⁢                      L                                                                                  )                                      ;                                                    and                                                                                                                            A                  11                                =                                                      -                                          ρ                      s                                                        ⁢                                      ω                    2                                    ⁢                                      ⅇ                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                                              k                        s                                            ⁢                      L                                                                                  ;                                                                                            A                  12                                =                                                      -                                          ρ                      s                                                        ⁢                                      ω                    2                                    ⁢                                      ⅇ                                          ⅈ                      ⁢                                                                                          ⁢                                              k                        s                                            ⁢                      L                                                                                  ;                                                                                            A                  13                                =                0                            ;                                                                                            A                  14                                =                                                      ⅈρ                    f                                    ⁢                  ω                                            ;                                                                                                            A                  21                                =                                                      -                                          ρ                      s                                                        ⁢                                      ω                    2                                                              ;                                                                                            A                  22                                =                                                      -                                          ρ                      s                                                        ⁢                                      ω                    2                                                              ;                                                                                            A                  23                                =                                                      -                                          ⅈρ                      f                                                        ⁢                  ω                                            ;                                                                                            A                  24                                =                0                            ;                                                                                                            A                  31                                =                                                      -                                          k                      s                                                        ⁢                                      ωⅇ                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                                              k                        s                                            ⁢                      L                                                                                  ;                                                                                            A                  32                                =                                                      k                    s                                    ⁢                                      ωⅇ                                          ⅈ                      ⁢                                                                                          ⁢                                              k                        s                                            ⁢                      L                                                                                  ;                                                                                            A                  33                                =                0                            ;                                                                                            A                  34                                =                                  ik                  f                                            ;                                                                                                            A                  41                                =                                                      -                                          k                      s                                                        ⁢                  ω                                            ;                                                                                            A                  42                                =                                                      k                    s                                    ⁢                  ω                                            ;                                                                                            A                  43                                =                                  ik                  f                                            ;                                                                          A                44                            =              0.                                                          (        25        )            
G7, the constant of interest for the fluid pressure field determination, can be solved as:
                                                        G              7                        =                                                                                (                                                                  H                        1                                            -                                                                                                    A                            11                                                    ⁢                                                      c                            ′                                                                                                    a                          ′                                                                                      )                                    ⁢                                      (                                                                  A                        32                                            -                                                                                                    A                            31                                                    ⁢                                                      b                            ′                                                                                                    a                          ′                                                                                      )                                                  -                                                      (                                                                  H                        3                                            -                                                                                                    A                            31                                                    ⁢                                                      c                            ′                                                                                                    a                          ′                                                                                      )                                    ⁢                                      (                                                                  A                        12                                            -                                                                                                    A                            11                                                    ⁢                                                      b                            ′                                                                                                    a                          ′                                                                                      )                                                                                                                    A                    14                                    ⁢                                      A                    32                                                  -                                                                            A                      14                                        ⁢                                          A                      31                                        ⁢                                          b                      ′                                                                            a                    ′                                                  -                                                      A                    34                                    ⁢                                      A                    12                                                  +                                                                            A                      34                                        ⁢                                          A                      11                                        ⁢                                          b                      ′                                                                            a                    ′                                                                                ,                                          ⁢          where                ⁢                                  ⁢                                            a              ′                        =                                                            A                  21                                ⁢                                  A                  43                                            -                                                A                  41                                ⁢                                  A                  23                                                              ;                ⁢                                  ⁢                                            b              ′                        =                                                            A                  22                                ⁢                                  A                  43                                            -                                                A                  42                                ⁢                                  A                  23                                                              ;                ⁢                                  ⁢                              c            ′                    =                                                    A                43                            ⁢                              H                2                                      -                                          A                23                            ⁢                                                H                  4                                .                                                                        (        26        )            
The pressure field can then be calculated as:P(z,ω)=iωρfG7eikf(z+L).   (27)
Equation (27) can be used to compute the PTA spectrum within a pre-selected range of modulation frequencies then using inverse Fourier transformation to reproduce the time-domain impulse response. Practical implementation of the PTA technique requires scanning the harmonic modulation frequency over the specified range and may be time-consuming for a wide frequency range. Use of frequency-swept (chirped) modulation signals and heterodyne detection provide a valuable alternative which allows recovery of the PTA spectrum and calculation of the temporal response. Taking into account the time-domain representation of harmonic signals in the form P(z,t)=P(z,ω) exp (iωt), the acoustic pressure field (27) at the transducer position z=−L −d (where d is the distance between the sample top surface and transducer, i.e. the sample depth) can be written as:P(t)|z=−L−d=iωρfG7eiω(t−d/cf)   (28)
When a laser source is modulated by a linear chirp with frequency sweep given by ω(t)=a+bt, where a−initial frequency and b is the sweep rate, the detected pressure wave is:P(t)=−i(a+bt)ρf|G7ei[(a+b(t−d/cf))t−a(d/ef)+θ]  (29)where the phase θ is due to the complex valued coefficient G7. The heterodyne signal can be generated by mixing the recorded PTA response with the frequency-swept modulation waveform and suppressing the high-frequency components using a low-pass filter. The resulting product term is:V(t)=P(t)·e−i[(a+bt)t]=−i(a+bt)ρf|G7|e−i[(bd/cf)t+ad/cf−θ)   (30)
It follows from equation (30), that the resulting heterodyne response is a harmonic signal oscillating at the frequency f=bd/2πcf, which depends on the sample depth d. Furthermore, it carries amplitude and phase modulation due to the dependence of the coefficient G7 on frequency ω(t). Equation (30) relates the spectral content of the heterodyne signal and the depth position of subsurface chromophores generating PTA waves. To improve the SNR of the PTA technique, the heterodyne signal can be measured using a suitable coherent (lock-in) processing algorithm which can be implemented by introducing a variable delay time τ to the chirped reference waveform and integrating the heterodyne signal over the entire acquisition time interval T:
                              V          ⁡                      (            τ            )                          =                                            ∫              T                        ⁢                                                            P                  ⁡                                      (                    t                    )                                                  ·                                  ⅇ                                      -                                          ⅈ                      ⁡                                              [                                                                                                            (                                                              a                                +                                                                  b                                  ⁡                                                                      (                                                                          t                                      -                                      τ                                                                        )                                                                                                                              )                                                        ⁢                            t                                                    -                                                      a                            ⁢                                                                                                                  ⁢                            τ                                                                          ]                                                                                                        ⁢                                                          ⁢                              ⅆ                t                                              =                                    -              ⅈ                        ⁢                                          ∫                T                            ⁢                                                ω                  ⁡                                      (                    t                    )                                                  ⁢                                  ρ                  f                                ⁢                                                                        G                    7                                                                    ⁢                                  ⅇ                                      ⅈ                    [                                                                                            b                          ⁡                                                      (                                                          τ                              -                                                              d                                /                                                                  c                                  f                                                                                                                      )                                                                          ⁢                        t                                            +                                              a                        ⁡                                                  (                                                      τ                            -                                                          d                              /                                                              c                                f                                                                                                              )                                                                    +                      θ                                        )                                                  ⁢                                                                  ⁢                                  ⅆ                  t                                                                                        (        31        )            
Equation (31) indicates that the heterodyne response will differ from zero only when the delay time τ is equal to the time d/cf, which is the time required for the acoustic waves to propagate a distance d.
In order to uncouple amplitude and phase of the heterodyne signal, the standard quadrature processing algorithm can be applied. It requires two reference waveforms with relative phase shift of π/2. Then heterodyne signals are formed by mixing both waveforms with the PTA response to determine in-phase V1(τ) and out-of-phase V2(τ) components according to equation (31). The resulting amplitude A and phase θ as functions of the delay time r can be determined as:
                                          A            ⁡                          (              τ              )                                =                                    (                                                                    V                    1                    2                                    ⁡                                      (                    τ                    )                                                  +                                                      V                    2                    2                                    ⁡                                      (                    τ                    )                                                              )                                      1              /              2                                      ⁢                                  ⁢                              θ            ⁡                          (              τ              )                                =                                    tg                              -                1                                      ⁡                          (                                                                    V                    2                                    ⁡                                      (                    τ                    )                                                                                        V                    1                                    ⁡                                      (                    τ                    )                                                              )                                                          (        32        )            Numerical Results
Theoretical simulations were performed for the simple case of a solid turbid layer immersed in water. Three input parameters, the optical absorption coefficient, optical scattering coefficient and the thickness of the solid were varied independently for each simulation to illustrate the time-domain PTA signal generation through the developed theory. Table 1 ( see D. P. Almond and P. M. Patel in Photothermal Science and Techniques, Chapman and Hall, 1996 and J. Krautkramer and H. Krautkramer in Ultrasonic Testing of Materials, Springer Verlag, 3rd ed., 1983) presents the optical and elastic properties used as input parameters for the mathematical model.
TABLE 1Elastic properties used as input parametersfor the numerical simulationρs (kg/m3)Ks (N/m2)βs (1/C)cs (m/s)cf (m/s)10000.5 × 1033.3 × 10−410001500ρf (kg/m3)λs (W/mK)λf (W/mK)αs (m2/s)αf (m2/s)9980.550.610.12 × 10−60.1 × 10−6
Equation (27) was used to calculate the laser-induced acoustic field within a user-selected frequency range. The time-domain results were obtained from their frequency-domain counterparts using Inverse Fourier Transformation (IFT). FIGS. 2(a), (b) and (c) illustrate the effects of varying optical absorption (penetration) depth on the PTA signal from a 5-mm thick turbid layer. The two peaks in each plot indicate the acoustic waves generated at the top and bottom surfaces of the turbid layer. It can be observed from these figures that increasing the optical absorption coefficient always results in an increase of the signal amplitudes corresponding to the first peaks. The magnitude of the second peak, however, is affected by the amount of energy transmitted to the bottom of the turbid layer and the amount of energy absorbed by that area. The magnitudes of the peaks in FIG. 2(b) are much larger than those in FIG. 2(a), due to an increase of the optical absorption coefficient, from 0.1 cm−1 to 1 cm−1. However, FIG. 2(c), which features an optical absorption coefficient of 4 cm−1, shows a degraded ratio of the peak magnitudes (second peak/first peak) compare to those of FIGS. 2(a) and 2(b). This indicates that although the optical absorption is higher in FIG. 2(c), significant amount of laser energy is absorbed during the light transmission process, resulting in a smaller amount of optical fluence being available to reach the back surface of the solid.
FIG. 3(a)-(d) feature the same material properties, but with an increased optical scattering coefficient, μs, from 0.5 cm−1 to 8 cm−1. This results in a significant decrease of the acoustic signals (peaks at around 46 μs) at the bottom surface of the turbid layer, which is due to the combined effects of energy absorption and scattering during light transmission. The PTA signals at the top surfaces (peaks at around 39 μs), however, show a slight increase as the scattering coefficient increases. This phenomenon, which is due to the localization of the optical source closer to the surface, is also evident in the experimental results shown in Section C) ii of DETAILED DESCRIPTION OF THE INVENTION.
FIG. 4(a)-(c) show results of numerical calculations of the PTA signal using quadrature detection technique (equations (31) and (32)). In this model, we assumed an optical absorption coefficient of 10 cm−1 of a 5 mm thick subsurface layer positioned 5 cm deep, in order to calculate in-phase and out-of-phase heterodyne signals (curves 1 and 2 respectively). Calculation of the amplitude (FIG. 4(b)) shows distinct peaks corresponding to acoustic waves generated at the sample surface and reflected back from the rear interface. The phase plot of the PTA signal shows dramatic changes when delay time corresponds to arrival of acoustic waves.